On the role of standard and anomalous diffusion in suspensions of rigid rods

Dilute suspensions composed of rods are usually described by using the Jeffery's model that only considers flow-induced orientation. When the concentration increases rods interaction cannot be neglected and the simplest way to take it into account is from a diffusion term that tends to recover an isotropic orientation distribution. However, the when considering standard diffusion the orientation kinematics seems to fast with respect to the experimental findings. Different approaches have been proposed by modifying diffison mechanisms in order to delay the orientation processes, however those approaches do not ptopsoe aphysical foundation related to the proposed mechanisms. In complex fluids, micro-rheological experiments often exhibit anomalous sub-diffusion or sticky diffusion, in which the mean square displacement of Brownian tracer particles is found to scale with a alpha power of the time, with alpha different of one (standard diffusion). In these cases, the use of non-integer derivatives can constitute an appealing alternative as it allows one to correctly reproduce the observed physical behaviour while keeping the model as simple as possible. Moreover, from a physical point of view, the use of non-integer derivatives introduces a degree of non-locality that seems in agreement with the intrinsic nature of the physical system. In the case of semi-dilute and semi-concentrated suspensions entanglements can create weakly interconnected networks responsible for the mild elasticity observed experimentally. In such a percolated system, Brownian motion is expected to be disturbed and to exhibit anomalous diffusion. In our previous wokrs we analyzed the effects of fractional diffusion in small amplitude oscillatory flows, proving that by considering a fractional derivative of order 0.5 it was possible reproduce a variety of experimental tests. This work constitutes a setp forward, in which the jsut described modelling framrwork is extended to nonlinear regimes, proivng that in this case the consideration of fractional diffusion allows delaying the flow induced orientation as experimenatlly noticed. In this work revisits the general framework and analyze the effects of using fractional derivatives on fdifferent types of flows, priving that is a valuable approach for modelling suspensions of particles.